# Abstract Nonsense

## The Irreps of the Product of Finitely Many Finite Groups

Point of post: In this post we shall discuss how one can find the set of all irreps, up to equivalence, of a group of the form $G_1\times\cdots\times G_n$ given the irreps of $G_k$ for $k\in[n]$.

Motivation

We’ve developed quite an extensive theory regarding how to find the irreps of a finite group $G$ and how to relate those irreps to one another, as well as their characters.The question remains though if there is a natural way that the irreps of $G$ relate naturally to constructions based on $G$. In particular, in this post we are interested in determining the relationships between the irreps (in particular the characters) of the product $G\times H$ of two groups $G$ and $H$ given the knowledge of the characters of $G$ and $H$. So for example, it’s easy (as we’ve shown) to construct the character table for $S_3$ and it’s equally easy to construct the character table for $\mathbb{Z}_2$. A next logical step would be to combine them and find the character table for $S_3\times\mathbb{Z}_2$. It would be nice if one would have to not go tromping through all the extra work to create this (larger) character table having gone through the (admittedly small amount of work) to construct the ones for $S_3$ and $\mathbb{Z}_2$. In this post we shall show that our greatest wish is true–we can’t just easily get some characters of $S_3\times\mathbb{Z}_2$ from those of $S_3$ and $\mathbb{Z}_2$ but we can easily get all of them.

April 11, 2011

## Character Table of S_3 By Finding the Irreducible Representations

Point of post: In this post we construct the first of a few character tables, namely we construct the character table for $S_3$.

Motivation

We now start off nice and easy and construct the classic character table for $S_3$ using the techniques from the last post. $S_3$ may be perhaps the easiest charater table to construct, but it will give us a good start to stretch our proverbial legs. In this post though, we find the character table using minimal machinery by actually constructing the irreducible characters of $S_3$ instead of using the techniques in our previous post. This method is, in my opinion, for the purpose of  character table construction, not preferable. Indeed, one must actually come up with representatives from each equivalency class of irreps of $S_3$. This post shall be useful to illustrate how beautifully simple the construction of character tables is made by the theory we’ve developed.

March 22, 2011

## Subrepresentations, Direct Sum of Representations, and Irreducible Representations (Pt. II)

Point of post: This post is a continuation of this one.

January 18, 2011